Journal of Global Optimization

, Volume 49, Issue 4, pp 623–649 | Cite as

Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices



We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate the problem of rounding a continuous solution to a mixed integer solution. A worst case complexity of a Lenstra-type algorithm is established using an approximate log-barrier center to obtain an ellipsoidal rounding of the feasible set. The results for the mixed integer convex programming also establish a complexity result for the mixed integer second order cone programming and mixed integer semidefinite programming feasibility problems as a special case. Our results motivate an alternative reformulation technique and a branching heuristic using a generalized (e.g., ellipsoidal) norm reduced basis available at the root node.


Linear programming Volumetric center Analytic center Interior point methods Convex programming Mixed integer programming Lattice basis reduction 


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  1. 1.
    Aardal, K., Hurkens, C., Lenstra, A.K.: Solving a Linear Diophantine Equation with Lower and Upper Bounds on the Variables. LNCS, vol. 1412, pp. 229–242 (1998)Google Scholar
  2. 2.
    Aardal K., Hurkens C.A.J., Lenstra A.K.: Solving a system of diophantine equation with lower and upper bounds on the variables. Math. Oper. Res. 25, 427–442 (2000)CrossRefGoogle Scholar
  3. 3.
    Aardal K., Bixby R., Hurkens C., Lenstra A.K., Smeltink J.: Market split and basis reduction: towards a solution of the Cornuéjols–Dawande instances. INFORMS J. Comput. 12, 192–202 (2000)CrossRefGoogle Scholar
  4. 4.
    Aardal K., Lenstra A.K.: Hard equality constrained integer knapsacks. Math. Oper. Res. 29(3), 724–738 (2004)CrossRefGoogle Scholar
  5. 5.
    Aardal K., Weismantel R., Wolsey L.A.: Non-standard approaches to integer programming. Discrete Applied Mathematics 123, 5–74 (2002)CrossRefGoogle Scholar
  6. 6.
    Atamtürk A., Narayanan V.: Cuts for Conic Mixed-Integer Programming. LCNS 4513, 16–29 (2007)Google Scholar
  7. 7.
    Anstreicher K.M.: Towards a practical volumetric cutting plane method for convex programming. SIAM J. Optim. 9(1), 190–206 (1998)CrossRefGoogle Scholar
  8. 8.
    Anstreicher K.M.: Ellipsoidal approximations of convex sets based on the volumetric barrier. Math. Oper. Res. 24(1), 193–203 (1999)CrossRefGoogle Scholar
  9. 9.
    Anstreicher, K.M.: Improved complexity for maximum volume inscribed ellipsoids, technical report. Departemnt of Management Science, University of Iowa, Iowa City (2001)Google Scholar
  10. 10.
    Babai L.: On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986)CrossRefGoogle Scholar
  11. 11.
    Bertsimas D., Tsitsiklis J.J.: Introduction to linear optimization. Athena Scientific (1997)Google Scholar
  12. 12.
    Bourgain J., Milman V.D.: New volume ratio properties for convex symmetric bodies in \({\mathbb R^n}\) . Inventiones Mathematicae 88, 319–340 (1987)CrossRefGoogle Scholar
  13. 13.
    Burago Y.D., Zalgaller V.A.: Geometric Inequalities. Springer, Berlin (1980)Google Scholar
  14. 14.
    Cassels J.W.S.: An introduction to the Geometry of Numbers. Springer, Berlin (1971)Google Scholar
  15. 15.
    CPLEX, CPLEX User’s Manual.
  16. 16.
    Cornuéjols, G., Liberti, L., Nannicini, G.: Improved strategies for branching on general disjunctions. Mathematical Programming A (to appear) (2010)Google Scholar
  17. 17.
    Cook W., Rutherford T., Scarf H.E., Shallcross D.: An implementation of the generalized basis reduction algorithm for integer programming. ORSA J Comput 5, 206–215 (1993)Google Scholar
  18. 18.
    Gao, L., Zhang, Y.: Computational Experience with Lenstra’s Algorithm, TR02-12, Department of Computational and Applied Mathematics, Rice University (2002)Google Scholar
  19. 19.
    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays, Presented to R. Courant on his 60th Birthday, pp. 187–204. Wiley, New York (1948)Google Scholar
  20. 20.
    Karamanov, M., Cornuéjols, G.: Branching on General Disjunctions. Mathematical Programming A (to appear) (2010)Google Scholar
  21. 21.
    Khachiyan L.G.: Rounding of polytopes in the real number model of computation. Math. Oper. Res. 21(2), 307–320 (1996)CrossRefGoogle Scholar
  22. 22.
    Khachiyan L.G., Todd M.J.: On the complexity of approximating the maximal volume inscribed ellipsoid for a polytope. Math. Program. 61, 137–159 (1993)CrossRefGoogle Scholar
  23. 23.
    Krishnamoorthy, B., Pataki, G.:Column basis reduction, and decomposable knapsack problems. Discrete Optimization (to appear) (2008)Google Scholar
  24. 24.
    Koy, H., Schnorr, C.P.: Segment LLL-reduction of lattice bases. In: Lecture Notes in Computer Science, vol. 2146, pp. 67–80 (2001)Google Scholar
  25. 25.
    Lenstra A.K., Lenstra H.W., Lovász L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)CrossRefGoogle Scholar
  26. 26.
    Lenstra H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)CrossRefGoogle Scholar
  27. 27.
    Lovász, L.: An algorithmic theory of numbers, Graphs and Convexity. SIAM (1986)Google Scholar
  28. 28.
    Lovász L., Lovász L., Lovász L.: The generalized basis reduction algorithm. Math. Oper. Res. 17(3), 751–764 (1992)CrossRefGoogle Scholar
  29. 29.
    Li Z., Mehrotra S.: An example to demonstrate the importance of using ellipsoidal norm in lattice basis reduction for branching on hyperplane algorithms. Pacific J. Optim. 5(2), 351–365 (2007)Google Scholar
  30. 30.
    Mahajan, A., Ralphs, T.K.: Experiments with branching using general disjunctions. In: Chinneck, J.W., Kristjansson, B., Saltzman, M.J. (eds.) Operations Research/Computer Science Interfaces (Chapter 6) (2009)Google Scholar
  31. 31.
    Mahajan, A., Ralphs, T.K.: On the complexity of selecting branching disjunctions in integer programming. Technical Report, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015 (2008)Google Scholar
  32. 32.
    Mehrotra, S., Li, Z.: Segment LLL reduction of latice bases using modular arithmetic. IE/MS technical report, Northwestern University, Evanston, IL 60208 (2001)Google Scholar
  33. 33.
    Mehrotra S., Ye Y.: Finding an interior point in the optimal face of linear programs. Math. Program. 62, 497–515 (1993)CrossRefGoogle Scholar
  34. 34.
    Nesterov Y.E., Nemirovski A.S.: Interior Point Polynomial Algorithms in Convex Programming. SIAM Publications, SIAM Philadelphia, USA (1994)Google Scholar
  35. 35.
    Owen J., Mehrotra S.: Experimental results on using general disjunctions in branch-and-bound for general-integer linear programs. Comp. Optim. Appl. 20(1), 159–170 (2001)CrossRefGoogle Scholar
  36. 36.
    Pardalos P.M., Ramana M.: Semidefinite programming. In: Terlaky, T. (eds) Interior Point methods of Mathematical Programming. pp. 369–398. Kluwer, Dordrecht (1996)Google Scholar
  37. 37.
    Pardalos P.M., Resende M.G.C.: Interior point methods for global optimization. In: Terlaky, T. (eds) Interior Point methods of Mathematical Programming, pp. 467–500. Kluwer, Dordrecht (1996)Google Scholar
  38. 38.
    Pardalos, P.M., Wolkowicz, H. (eds) Topics in semidefinite and interior-point methods. Fields Institute Communications Series, vol. 18, American Mathematical Society (1998)Google Scholar
  39. 39.
    Renegar J.: A mathematical view of interior-point methods in convex optimization. MPS-SIAM Ser. Optim. 40, 59–93 (2001)Google Scholar
  40. 40.
    Schrijver A.: Theory of Linear and Integer Programming. Wiley, New York (1986)Google Scholar
  41. 41.
    Tarasov S.P., Khachiyan L.G., Erlich I.I.: The method of inscribed ellipsoids. Soviet Math. Doklady 27, 226–230 (1988)Google Scholar
  42. 42.
    Vaidya P.M.: A new algorithm for minimizing convex functions over convex sets. Math. Program. 73, 291–341 (1996)Google Scholar
  43. 43.
    Wang, X.: An implementation of the generalized basis reduction algorithm for convex integer programming. Ph.D. dissertation, Department of Economics, Yale University (1997)Google Scholar
  44. 44.
    Ye Y.: On the convergence of interior-point algorithms for linear programming. Math. Program. 57, 325–335 (1992)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Management Sciences, Robert R. McCormick School of EngineeringNorthwestern UniversityEvanstonUSA

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